Tag Archives: determinant

Integer sets without singular matrices

Let’s say that a set A\subset \mathbb N contains an n\times n singular matrix if there exists an n\times n matrix with determinant zero whose entries are distinct elements of A. For example, the set of prime numbers does not contain any 2\times 2 singular matrices; however, for every n\ge 3 it contains infinitely many n\times n singular matrices.

I don’t know of an elementary proof of the latter fact. By a 1939 theorem of van der Corput, the set of primes contains infinitely many progressions of length 3. (Much more recently, Green and Tao replaced 3 with an arbitrary k.) If every row of a matrix begins with a 3-term arithmetic progression, the matrix is singular.

In the opposite direction, one may want to see an example of an infinite set A\subset \mathbb N that contains no singular matrices of any size. Here is one:
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